MATH 119 Section 5.3 : Finding Area Between Curves
Important: In this section we will learn just how to set up the integrals to find area between curves. The final answer for each example in this handout is given for information only. You will be required to solve same problems with final answers when we cover section 7.3.
A) Area Between Curves in [a , b]
Total Area between f(x) and g(x)in [a , b] / = / -
=
Example 1: Find the area of the region that is enclosed by , , x = -1 and x = 2.
Solution:
1) Graph the functions , and . Shade in the region in the interval [-1 , 2]
2) The function is on the top in the entire interval of [-1 , 2]
3) Area:
= 15
B) Area Between a Curve and the x-Axis in [a , b] (x-axis is same as y = 0)
=
= / Top function is y = 0, Bottom is y = f(x)
=
= or Area
Example 2: Find the area of the region that is enclosed by the x-axis, the graphover the interval [-1, 4].
Solution:
1) Graph the functions , and (the x-axis). Shade in the region in the interval [-1 , 4]
2) The function is on the top in the entire interval of [-1 , 4]
3) Area:
= 31.67
C) Area Between Intersecting Curves
When [a , b] is not given, then we must determine the appropriate values from the graph. The area bounded between two functions and is from the intersection point on the left to the intersection point on the right. To find these points of intersection, we solve the equation: for x.
Example 3: Find the area of the region that is enclosed by the graphs of the functions:
and
Solution:
1) Graph both functions
2) Find the points of intersection of the two curves by equating the two equations :
and :
Then x = -1 or x = 3
3) Area:
Example 4: Find the area of the region that is enclosed by and the x- axis.
Solution:
1) Graph the functions , and y = 0 (the x-axis)
2) Find the points of intersection by equating the two equations and :
Then x = 1 or x = 3
3) Area:
More Examples:
Example 5: Find the area of the region that is enclosed by the graphs of the functions:
and
Solution:
1) Graph both functions
2) Find the points of intersection of the two curves by equating the two equations :
and
Square both sides:
Then x = 0 or x = 1
3) Area:
Example 6: Find the area of the region that is enclosed by the graphs of the functions:
and
.
Solution:
1) Graph both functions
2) Find the points of intersection of the two curves by equating the two equations :
=
Then x = -2 or x = 2
3) Area:
21.33
D) Area Between Curves With Multiple Points of Intersection (Crossing Curves)
If neither graph or lies above the other over the whole interval, then we break the area into two pieces. One on either side of the point at which the graphs cross and then compute each area separately.
To do this, we need to know exactly where that crossing point is by solving the equation: for x.
Example 7: Find the area of the region that is enclosed by the x-axis, over the interval [-1, 3].
Solution:
1) Graph the functions , and y = 0 (the x-axis). Shade in the region in the interval [-1 , 3]
2) Find the points of intersection by equating the two equations and :
The graph crosses the x-axis at: x = -2 , x = 2
3) The total area is the sum of the two areas: [-1 , 2] and [2 , 3]
+
+
+ = 9 + 2.33 = 12.33
Note 1: If you write the area as , then the answer will be -9 + 2.33 = -6.67 .
Note 2: You should get the correct answer for the Total Area as long as each integral produces a positive area. This is one way to check your work.
Example 8: Find the area of the region that is enclosed by the x-axis, , x = -1 and x = 2.
Solution:
1) Graph and shade the region between x = -1 and x =2
2) Find the points of intersection:
or
The graph crosses the x-axis at x = -1 and x = 1
3) + = 1.33 + 1.33 = 2.66
Example 9: Find the area of the region that is enclosed by the graphs of the functions:
, , over the interval [-1 , 3].
Solution:
1) Graph the functions , and .
2) Find the points of intersection of the two curves by equating the two equations :
The two graphs cross when x = -1 and x = 2.
3) The total area is the sum of the two areas: [-1 , 2] and [2 , 3]
= 4.5 + 1.83 = 6.33
Example 10: Find the area of the region that is enclosed by the graphs of the functions:
,
Solution:
1) Graph the functions , and .
2) Find the points of intersection of the two curves by equating the two equations :
The two cross at: x = 0, x = -1or x = 1
3) The total area is the sum of the two areas: [-1 , 0] and [0 , 1]
+
+
= 0.25 + 0.25 = 0.5 (notice that the answer will be zero if one area is negative)
2